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Locally Realized Covering Relation

Let be a lattice (or a bounded lattice or a complemented lattice, etc.), and let be the covering relation of :

Then is locally realized provided that for every finite subset of , there is a finitely generated sublattice of , that contains , for which . It can be shown that is locally realized if and only if there is a hyperfinitely generated sublattice of such that . Using this characterization of locally realized covering relations, the following standard result can be proved using nonstandard methods:

Let be a locally finite lattice in which the covering relation is locally realized, and let be the sublattice of which is generated by . Then is a connected tolerance of , and it is in fact the smallest locally subconnected (and locally connected) tolerance of .

This entry contributed by Matt Insall (author's link)

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References

Burris, S. and Sankappanavar, H. P. A Course in Universal Algebra. New York: Springer-Verlag, 1981. http://www.thoralf.uwaterloo.ca/htdocs/ualg.html.Gehrke, M.; Kaiser, K.; and Insall, M. "Some Nonstandard Methods Applied to Distributive Lattices." Zeitschrifte für Mathematische Logik und Grundlagen der Mathematik 36, 123-131, 1990.Grätzer, G. Lattice Theory: First Concepts and Distributive Lattices. San Francisco, CA: W. H. Freeman, 1971.Grätzer, G. Universal Algebra, 2nd ed. New York: Springer-Verlag, 1979.Insall, M. "Some Finiteness Conditions in Lattices Using Nonstandard Proof Methods." J. Austral. Math. Soc. 53, 266-280, 1992.

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Locally Realized Covering Relation

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Insall, Matt. "Locally Realized Covering Relation." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/LocallyRealizedCoveringRelation.html